## Abstract

We considered theoretically the nonlinear interaction of surface plasmon polaritons (SPPs) in a metal-insulator-metal (MIM) plasmonic waveguide with a smectic liquid crystalline core. The interaction is related to the specific cubic optical nonlinearity mechanism caused by smectic layer oscillations in the SPP electric field. The interfering SPPs create the localized dynamic grating of the smectic layer strain that results in the strong stimulated scattering of SPP modes in the MIM waveguide. We solved simultaneously the smectic layer equation of motion in the SPP electric field and the Maxwell equations for the interacting SPPs. We evaluated the SPP mode slowly varying amplitudes (SVAs), the smectic layer dynamic grating amplitude, and the hydrodynamic velocity of the flow in a smectic A liquid crystal (SmALC).

### Keywords

- surface plasmon polariton (SPP)
- smectic liquid crystals
- stimulated light scattering (SLS)
- plasmonic waveguide

## 1. Introduction

Nonlinear optical phenomena based on the second- and third-order optical nonlinearity characterized by susceptibilities

Nonlinear optical effects can be enhanced by plasmonic excitations as follows: (i) the coupling of light to surface plasmons results in strong local electromagnetic fields; (ii) typically, plasmonic excitations are highly sensitive to dielectric properties of the metal and surrounding medium [1]. In nonlinear optical phenomena, such a sensitivity can be used for the light-induced nonlinear change in the dielectric properties of one of the materials which result in the varying of the plasmonic resonances and the signal beam propagation conditions [1]. Plasmonic excitations are characterized by timescale of several femtoseconds which permits the ultrafast optical signal processing [1]. The SPP field confinement and enhancement can be changed by modifying the structure of the metal or the dielectric near the interface [1]. For example, plasmonic waveguides can be created [1, 7, 8, 9]. Nanoplasmonic waveguides can confine and enhance electric fields near the nanometallic surfaces due to the propagating SPPs [9]. Nanoplasmonic waveguide consists of one or two metal films combined with one or two dielectric slabs [9]. Typically, two types of the plasmonic waveguides exist: (i) an insulator/metal/insulator (IMI) heterostructure where a thin metallic layer is placed between two infinitely thick dielectric claddings and (ii) a metal/insulator/metal (MIM) heterostructure where a thin dielectric layer is sandwiched between two metallic claddings [7]. The MIM waveguides for nonlinear optical applications require highly nonlinear dielectrics [9]. The nonlinear metamaterials can significantly increase the nonlinearity magnitude [10]. Investigation of nonlinear metamaterials is related in particular to nonlinear plasmonics and active media [10]. One of the metamaterial nonlinearity mechanisms is based on liquid crystals (LCs) [10]. Tunability and a strongly nonlinear response of metamaterials can be obtained by their integration with LCs offering a practical solution for controlling metamaterial devices [11].

The integration of LCs with plasmonic and metamaterials may be promising for applications in modern photonics due to the extremely large optical nonlinearity of LCs, strong localized electric fields of surface plasmon polaritons (SPPs), and high operation rates as compared to conventional electro-optic devices [12]. Practically all nonlinear optical processes such as wave mixing, self-focusing, self-guiding, optical bistabilities and instabilities, phase conjugation, SLS, optical limiting, interface switching, beam combining, and self-starting laser oscillations have been observed in LCs [13]. LC can be incorporated into nano- and microstructures such as a MIM plasmonic waveguide. Nematic LCs (NLCs) characterized by the orientation long-range order of the elongated molecules are mainly used in optical applications including plasmonics and nanophotonics [11, 12, 13, 14]. For instance, light-induced control of fishnet metamaterials infiltrated with NLCs was demonstrated experimentally where a metal-dielectric (Au-MgF_{2}) sandwich nanostructure on a glass substrate with the inserted NLC was used [11]. However, the NLC applications are limited by their large losses and relatively slow response [14, 15]. The light scattering in smectic A LC (SmALC) waveguides had been studied theoretically and experimentally, and it was shown that the scattering losses in SmALC are much lower than in NLC due to a higher degree of the long-range order [15]. SmALC can be useful in nonlinear optical applications and low-loss active waveguide devices for integrated optics [14, 15].

SmALCs are characterized by a positional long-range order in the direction of the elongated molecular axis and demonstrate a layer structure with a layer thickness

The nonlinear optical phenomena in SmALC such as a light self-focusing, self-trapping, SPM, SLS, and FWM based on the specific mechanism of the third-order optical nonlinearity related to the smectic layer normal displacement had been investigated theoretically [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In particular it has been shown that at the interface of a metal and SmALC, the counter-propagating SPPs created the dynamic grating of the smectic layer normal displacement

In this chapter we consider theoretically the interaction of the counter-propagating SPP modes in the MIM waveguide with the SmALC core. The interfering SPP TM modes with the close optical frequencies

The chapter is constructed as follows. The hydrodynamics of SmALC in the external electric field is considered in Section 2. The SPP modes of the MIM waveguide are derived in Section 3. The SPP SVAs, the smectic layer dynamic grating amplitude, and the SmALC hydrodynamic velocity are evaluated in Section 4. The conclusions are presented in Section 5.

## 2. Hydrodynamics of SmALC in the external electric field

In this section we briefly discuss the SmALC hydrodynamics and derive the equation of motion for the smectic layer normal displacement

Smectic layer oscillations

Here, * F*in the presence of the external electric field

Here * F*(7) can be neglected since for the typical values of the elastic constants

where

Here

Here, _{0} are the SS frequency and velocity, respectively [29]. It is seen from Eq. (10) that the SS frequency

If the viscosity terms responsible for the SS wave decay can be neglected, then the homogeneous part of Eq. (9) reduces to the SS wave equation with the dispersion relation (10) [29, 30, 31]:

We use equation of motion (9) for the evaluation of the light-enhanced dynamic grating

## 3. SPP modes in a MIM waveguide with SmALC core

LC slab optical waveguide represents a LC layer of a thickness about 1 μm confined between two glass slides of lower refractive index than LC [14]. LC as a waveguide core provides the photonic signal modulation and switching by using the electro-optic or nonlinear optical effects of LC mesophases [35]. For instance, the large optical nonlinearities were implemented in order to create optical paths by photonic control of solitons in NLC [35]. Various electrode geometries may create due to the electro-optic effect periodically modulated LC core waveguides which can serve as efficient guided distributed Bragg reflectors with the tuning ranges of about 100–1550 nm optical wavelength range [35]. Plasmonic waveguides based on the manipulation and routing of SPPs can demonstrate a subwavelength beyond the diffraction limit together with large bandwidth and high operation rate typical for photonics [36]. The plasmonic devices can be integrated into nanophotonic chips due to their small scale and the compatibility with the VLSI electronic technology [36]. Plasmonic devices are the promising candidates for future integrated photonic circuits for broadband light routing, switching, and interconnecting [36]. It has been shown that different plasmonic structures can provide SPP light waveguiding determining the SPP mode properties [36]. MIM waveguide representing a dielectric sandwiched between two metal slabs attracted a research interest as a basic component of nanoscale plasmonic integrated circuits [37]. LC-tunable waveguides have been proposed as a core element of low-power variable attenuators, phase-shifters, switches, filters, tunable lenses, beam steers, and modulators [37, 38]. Typically NLCs have been used due to their strong optical anisotropy, responsivity to external electric and magnetic fields, and low power [37, 38]. Different types of NLC plasmonic waveguides have been proposed and investigated theoretically [36, 37, 38]. Recently, SmALCs attracted attention due to their layered structure and reconfigurable layer curvature [39]. The possibility of the dynamic variation of smectic layer configuration by external fields is intensively studied [39]. We investigated theoretically SLS in the optical slab waveguide with the SmALC core where the third-order optical nonlinearity mechanism was related to the smectic layer dynamic grating created by the interfering waveguide modes [27]. We also considered theoretically the MIM waveguide with the SmALC core [24, 26].

The structure of such a symmetric waveguide of the thickness 2d is shown in Figure 3 [24, 26]. The plane of the waveguide is perpendicular to the SmALC optical axis * Z*. The SmALC in the waveguide core is homeotropically oriented, i.e., the smectic layers are parallel to the waveguide claddings

*axis [29]. Typically the waveguide dimension in the*Z

*axis direction is much larger than*Y

*, and the dependence on the coordinate*d

*in Eqs. (8) and (9) can be omitted. Than we obtain*y

The permittivity

where _{0} is the free electron density in the metal;* e*,

*are the electron charge and mass, respectively; and*m

Here _{0} is the free space permeability and

The linear part

Here c.c. stands for complex conjugate. The SPP fields (17)–(20) are confined in the * Z*direction. In the linear approximation substituting expressions (15), (18), and (20) into the homogeneous part of the wave equation (14) for the claddings and SmALC core, respectively, we obtain the following expressions for the complex wave numbers

where * c*is the free space light velocity. The boundary conditions for the fields (17)–(20) at the interfaces

Substituting expressions (17)–(20) into Eqs. (23) and (24), we obtain the dispersion relation for the SPP TM modes in the MIM waveguide given by [24, 26]

Dispersion relation obtained for the general case of different claddings [7] coincides with expression (25) for the symmetric structure with the same claddings. The results of the numerical solution of Eq. (25) for the typical values of the MIM waveguide parameters and the SPP frequencies

These results show that

The numerical estimations show that for the SPP modes with the close optical frequencies

Substituting expression (27) into equation of motion (9), we obtain the expression of the smectic layer displacement localized dynamic grating

Here

Expression (28) is the enhanced solution of Eq. (9). The homogeneous solution of Eq. (9) is overdamped for the typical values of SmALC parameters and * Z*direction and oscillates in the propagation direction

*.*X

## 4. Nonlinear interaction of SPPs in the MIM waveguide

The light-enhanced dynamic grating (28) results in the nonlinear polarization defined by Eq. (16). In order to investigate the interaction of the counter-propagating SPPs (27), we should solve wave Eq. (14) including the nonlinear term * x*coordinate can be neglected. We assume according to the SVA approximation that

Substituting expressions (27) and (28) into Eqs. (16), we evaluate the nonlinear part

Here we assumed that the factor

Here we neglected the small quantities

where the localization factor

The spectral dependence of the localization factor

It is seen from Figure 7 that

We obtain from Eq. (39) the Manley-Rowe relation for the SVA magnitudes

We introduce the dimensionless quantities

Substituting relationship (41) into Eq. (36), we obtain

where the gain * g*has the form

The spectral dependence of the gain * g*is shown in Figure 8. The solution of Eq. (41) has the form

It is easy to see from Eqs. (44) and (45) that the solutions _{1} with the normalized intensity

It is seen form Figure 6 that for

Substitute expression (46) into Eqs. (44) and (45). Then they take the form

The time duration of the energy exchange between the SPPs is about

The temporal dependence of the SPP SVA phases

It is seen from expressions (48) and (49) that SLS of the SPPs in the MIM waveguide is accompanied by XPM. For the large time intervals

Such a behavior corresponds to the rapid oscillations of the depleted pumping wave amplitude. The signal wave phase for

Substituting expressions (41) and (47) into Eq. (29), we obtain the explicit expression for the dynamic grating amplitude. It takes the form

The temporal dependence of the amplitude (52) normalized absolute value

We evaluate now the hydrodynamic flow velocity in the MIM wave guide core. Substituting expression (28) into Eqs. (1) and (6), we obtain

Expressions (28) and (52)–(55) and Figure 11 show that the orientational and hydrodynamic excitations in SmALC core of the MIM waveguide enhanced by the SPPs are spatially localized and reach their maximum value during the time of the energy exchange between the interacting SPPs.

## 5. Conclusions

We investigated theoretically the nonlinear interaction of SPPs in the MIM waveguide with the SmALC core. The third-order nonlinearity mechanism is related to the smectic layer oscillations that take place without the change of the mass density. We solved simultaneously the equation of motion for the smectic layer normal displacement and the Maxwell equations for SPPs including the nonlinear polarization caused by the smectic layer strain. We evaluated the dynamic grating of the smectic layer displacement enhanced by the interfering SPPs. We evaluated the SVAs of the interacting SPPs. It has been shown that the SLS of the orientational type takes place. The pumping wave is depleted, while the signal wave is amplified up to the saturation level defined by the total intensity of the interacting waves. SLS is accompanied by XPM. The phase of the depleted pumping wave rapidly increases, while the phase of the amplified wave tends to a constant value. The SPP characteristic rise time is of the magnitude of 10^{−9} s for a feasible SPP electric field of 10^{6} V/m. The smectic layer displacement and hydrodynamic velocity enhanced by SPPs are spatially localized and reach their maximum value during the time of the strong energy exchange between the interfering SPPs.